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Topic: On the Number of Primes Less Than a Given Magnitude

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Riemann hypothesis

Riemann zeta function

Riemann integral

Riemann surface

Extendible cardinal

Ramsey cardinal

Bernhard Riemann

List of important publications in Mathematics

Subtle cardinal

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Integral

Algebraic geometry

Grothendieck topology

Huge cardinal

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	Basel problem - Wikipedia, the free encyclopedia
		The Basel problem is a famous problem in number theory, first posed by Pietro Mengoli in 1644, and solved by Leonhard Euler in 1735.
		Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties.
		The Riemann zeta function ζ(s) is one of the most important functions in mathematics, because of its relationship to the distribution of the prime numbers.
	en.wikipedia.org /wiki/Basel_problem (1511 words)

	List of publications in mathematics - Encyclopedia, History, Geography and Biography
		Description: On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy.
		Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens of researchers during the late 19th century and up to the present day.
		Description: Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863.The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert.
	www.arikah.net /encyclopedia/List_of_publications_in_mathematics (4128 words)

	List of publications in mathematics - Wikipedia, the free encyclopedia
		More than any specific result in the publication, it seems that the major achievement of this publication is the popularization of logic and mathematical proof as a method of solving problems.
		Description: Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863.
		The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert.
	en.wikipedia.org /wiki/List_of_important_publications_in_mathematics (4128 words)

	hwaien
		In the 1970s, the study of prime numbers was suddenly pulled out of the ivory towers of pure mathematics and into the rough and dirty world of commerce.
		The hope was that by constructing formulae describing specific primes, one might eventually stumble upon the secret to constructing a formula describing prime numbers in general.
		In 1801, Gauss proposed that the number of primes distributed between 1 and n is roughly equal to n over the natural logarithm of n.
	hwaien.blogspot.com (9846 words)

	Riemann hypothesis
		Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he did not attempt a proof.
		The zeta function has a deep connection to the distribution of prime numbers and Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the following considerable strengthening of the prime number theorem:
		The zeros of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.
	www.askfactmaster.com /RH (1039 words)

	Gresham College \| Transcript
		Prime numbers are central to mathematics because they form the building blocks for numbers – every whole number can be built up from them: they’re the ‘atoms’ or ‘fundamental particles’ of mathematics.
		First, there’s a number x, which is about 1.3064, with the property that if you raise it to the powers 3, 9, 27, 81 (the powers of 3), and then drop the fractional bits, you always get a prime number.
		The first is that the prime numbers belong to the most arbitrary objects studied by mathematicians: they grow like weeds, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.
	www.gresham.ac.uk /printtranscript.asp?EventId=275 (4106 words)

	Number theory algorithms
		The probability for a given number n to be prime is roughly p[p]=1/Ln(n) and to be composite p[c]=1-1/Ln(n).
		After separating small prime factors, we test whether the number n is an integer power of a prime number, i.e.
		Let p[1]^k[1]...p[r]^k[r] be the prime factorization of n, where r is the number of prime factors and k[r] is the multiplicity of the r-th factor.
	yacas.sourceforge.net /Algochapter2.html (3917 words)

	math lessons - Riemann hypothesis
		One is the rate of growth in the error term of the prime number theorem given above.
		Another conjecture is the large prime gap conjecture; Cramér proved that on the Riemann hypothesis we have that the largest gaps between successive prime numbers is
		Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire.
	www.mathdaily.com /lessons/RH (1391 words)

	Riemann’s Hypothesis
		Numbers that are not prime numbers are composite numbers, excepting 1, which is the unit.
		The number of primes is infinite (Euclid originally proved this, and there are several alternative proofs), but this does not tell us whether the number of primes increases, remains the same, or decreases with x.
		In 1914 Hardy proved that there are an infinite number of zeroes on the critical line, Hardy and Littlewood established that there are at least KT zeroes on the critical line (K constant, T is the height being considered), and Selberg improved this to KTlogT.
	www.maths.ex.ac.uk /~mwatkins/zeta/perryRHproof.htm (1155 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		The number pi(N) of primes less than N is about N/log(N) (with an error on the order of O(N/(logN)^2) so the number of primes in one of these intervals is about 10^k/(klog(10) + log(a)) (with an error on the order of 10^k/k^2).
		It follows that when counting the primes in intervals of a given length, the number of primes is smaller when the starting point A is larger.
		The number of primes between 10^r and 10^s (r, s need not be integral) is about \int_r^s 10^u / u du In order to count the primes between 10 x 10^k and 99 x 10^k with a given second digit, simply evaluate this integral with r = k + log_10(10+i), s = k + log_10(11+i)...
	www.math.niu.edu /~rusin/known-math/01_incoming/2nd_digit (565 words)

	The 1896 Proof of Prime Number Theorem
		Possibly the single most important problem in Analytic Number Theory, and certainly the driving force behind many of its' results, is the question of the distribution of the prime numbers.
		The proof, at the end of the nineteenth century, of what is now known as the Prime Number Theorem is one of mathematics' greatest achievements, representing the culmination of the work of some of history's finest mathematicians.
		The distribution of the prime numbers, like so many other areas of mathematics, was first investigated by Euclid, who proved that the number of primes is infinite.
	www.freewebs.com /history_of_mathematics (991 words)

	Open Questions: The Riemann Hypothesis
		Intuitively, the reason that primes become sparser as they grow larger is that in some sense it becomes more "difficult" for a large number to be prime the larger it is. This is because there are an ever increasing number of smaller numbers (primes among them) which could divide it.
		Given that there are in fact two product formulas for ζ(s), that one involves the prime numbers, and that the other involves the zeros of ζ(s), it's hard to avoid the suspicion that there is some relationship between the distribution of the primes and the distribution of the zeros.
		The prime number theorem, the Riemann hypothesis, the functional equation of the zeta function, and the abc conjecture are the topics of interest with respect to number theory.
	www.openquestions.com /oq-ma014.htm (14106 words)

	Sandbox
		Or, that one of the greatest mathematicians of all times and the first number theorist was Pierre de Fermat who was an amateur mathematician, while professionally, he was a lawyer and jurist.
		The odd numbered chapters are mathematical and the even numbered ones are biographical and historical.
		In his 1859 paper On the Number of Primes Less Than a Given Magnitude, Bernhard Riemann (1826-1866) examined the properties of the latter mentioned relationship.
	www.hi-tech21.com /sandbox.htm (1471 words)

	Factorization using the Elliptic Curve Method
		The final value must have 10000 or less digits, intermediate results must have 20000 or less digits and in the case of divisions, the dividend must be multiple of the divisor.
		The next table shows the optimal values of B1 given the number of digits of the factor and the expected number of curves using that limit.
		When the number to be factorized is in the range 31-90 digits, after computing some curves in order to find small factors, the program switches to SIQS (if the checkbox located below the applet enables it), which is an algorithm that is much faster than ECM when the number has two large prime factors.
	www.alpertron.com.ar /ECM.HTM (618 words)

	BERNHARD RIEMANN
		This seemingly esoteric condition is of fundamental importance for the distribution of prime numbers.
		Riemann's lecture, "On the hypotheses that lie at the foundation of geometry" was given on June 10, 1854.
		His report, titled "On the number of primes less than a given magnitude", was of fundamental importance in number theory.
	www.usna.edu /Users/math/meh/riemann.html (1057 words)

	Riemann, Georg Friedrich Bernhard Riemann - Famous mathematicians pictures, posters, gifts items, note cards, greeting ...
		Riemann essentially increased the number of 'x's", so that there were enough for each possible y.
		In his 1859 paper On the Number of Primes Less Than a Given Magnitude, Reimann dealt with a formula for the identification and track the occurrence of prime numbers (a prime number being one which has no factor except itsef and 1).
		Prime numbers not only have an almost mystical fascination, but seem to hold the keys to many perplexing riddles.
	www.mathematicianspictures.com /Mathematicians/Riemann.htm (396 words)

	Non-deductive Logic in Mathematics
		It is recognised that of the enormous number of unsolved problems that have been or could be thought of, the tractable ones form a small proportion, and that it is difficult to discern which they are.
		What concern there was was less about the completion of the project than about what to do next; the editor of the conference proceedings began by commenting, ‘In the last year or so there have been widespread rumors that group theory is finished, that there is nothing more to be done’ (Mason [1980], p.
		Riemann, B. [1859]: ‘On the Number of Primes Less Than a Given Magnitude’, translated in Edwards [1974], pp.
	web.maths.unsw.edu.au /~jim/nondeductivelogic.html (7220 words)

	Earth & Sky : Radio Shows (Site not responding. Last check: 2007-10-22)
		Prime numbers can only be evenly divided by themselves and one.
		For example, 3 is a prime number because you can only divide it evenly by 3 and 1.
		The first few prime numbers are: 2, 3, 5, 7, 11, 13 and so on.
	www.earthsky.com /shows/show.php?date=20050128 (395 words)

	Amazon.com: Riemann's Zeta Function (Pure and Applied Mathematics; a Series of Monographs and Textbooks, 58): Books: ... (Site not responding. Last check: 2007-10-22)
		It includes a translation of Riemann's original paper (On the Number of Primes...) which is very nice and most authors now seem to forget to mention (mainly because of the obscure way in which it was written).
		The fifth one includes an error estimation due to Poussin for the prime number theorem, and the equivalent of the Riemann Hypothesis in terms of prime distributions.
		Riemann feels that all nontrivial zeros have real part 1/2, but this doesn't really matter right now since the term in the prime density expression depending on the zeros is "periodic" in any case and Riemann thus discards it without much harm when he derives his expression for the number of primes less than x.
	www.amazon.com /exec/obidos/tg/detail/-/0122327500?v=glance (2030 words)

	Riemann Hypothesis in a Nutshell (Site not responding. Last check: 2007-10-22)
		In his 1859 paper On the Number of Primes Less Than a Given Magnitude, Bernhard Riemann (1826-1866) examined the properties of the function
		less than or equal to zero the zeta function has zeros at the negative even integers; these are known as the trivial zeros.
		is the number of roots in the critical strip between zero and
	web.mala.bc.ca /pughg/RiemannZeta/RiemannZetaLong.html (1384 words)

	Riemann's 1859 Paper
		This theorem, first conjectured by Gauss when he was a young man, states that the number of primes less than x is asymptotic to x/log(x).
		To draw a statistical analogy, if the prime number theorem tells us something about the average distribution of the primes along the number line, then the Riemann hypothesis tells us something about the deviation from the average.
		The Riemann hypothesis was one of the famous Hilbert problems — number eight of twenty-three.
	www.claymath.org /millennium/Riemann_Hypothesis/1859_manuscript (437 words)

	[No title]
		The word "easily" means that using the data to prove primality is much easier and faster than generating the data in the first place.
		Multiplying the numbers together to show that they are the factors is much easier than finding the factors.
		These certificates are based on showing that simple properties that are true of prime numbers are not true for the given number.
	www.math.niu.edu /~rusin/known-math/96/porbable.prime (733 words)

	Number Theory
		Two classic essays by great German mathematician: on the theory of irrational numbers; and on transfinite numbers and properties of natural numbers.
		Gausian primes; polynomials over a field; algebraic number fields; algebraic integers and integral bases; uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal c...
		Superb, high-level study of landmark 1859 publication entitled "On the Number of Primes Less Than a Given Magnitude" traces developments in mathematical theory that it inspired.
	store.doverpublications.com /0486682528.html (417 words)

	Riemann Hypothesis - Page 2 (Site not responding. Last check: 2007-10-22)
		now the problem riemann considered was that of determining roughly the number of prime numbers that are less than a given number x = pi(x).
		Then Gauss conjectured that the number pi(x) was well approximated by Li(x) = the integral from 0 to x of dt/ln(t), (up to a small constant).
		not just the number of primes less than x but also 1/2 the number of squares of primes, plus 1/3 the number of cubes of primes,.....
	www.physicsforums.com /showthread.php?t=78799&page=2 (777 words)

	Riemann
		In the spring of 1846 Riemann enrolled at the University of Göttingen.
		A newly elected member of the Berlin Academy of Sciences had to report on their most recent research and Riemann sent a report on On the number of primes less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way.
		Here the sum is over all natural numbers n while the product is over all prime numbers.
	www.meta-religion.com /Mathematics/Biography/riemann.htm (2629 words)

	Soy Candles Store: Soy Candles - Online Cheap Soy Candles
		less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way.
		Soy Candles considered a very different question to the one Euler had considered, for he looked at the zeta function as a complex function rather than a real one.
		The main purpose of the paper was to give estimates for the number of primes less than a given number.
	www.cheap-candles-store.com /soy-candles.html (2744 words)

	Background on the number field sieve
		Current PC implementations intended for the factorization of relatively small numbers usually have adequate memory for sieving.
		Thus, despite the fact that current implementations of the relation collection require substantial memory, it is well known that asymptotically this step requires negligible memory without incurring, in theory, a runtime penalty - in practice, however, it is substantially slower than sieving.
		This corresponds to the heuristic asymptotic runtime of the NFS as given in [
	www.wisdom.weizmann.ac.il /~tromer/papers/meshc/node2.html (985 words)

	Category:Analytic number theory - Wikipedia, the free encyclopedia
		Analytic number theory is the branch of number theory that uses methods from mathematical analysis.
		Its first major success was the application of complex analysis in the proofs of the prime number theorem based on the Riemann zeta function.
		There are 3 subcategories shown below (more may be shown on subsequent pages).
	en.wikipedia.org /wiki/Category:Analytic_number_theory (112 words)

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